chore: add an example app for browser sdk

[joker23]

This PR will add a new browser SDK example that works with the new 4.x implementation.

---

Note

Adds a new browser example app demonstrating the JS client SDK, with build config and workspace inclusion.

• Example app (packages/sdk/browser/example):
  • App logic: src/app.ts initializes @launchdarkly/js-client-sdk, identifies a user context, evaluates flagKey, updates UI on change/error.
  • Build/setup: package.json with tsdown build, tsconfig.json, tsdown.config.ts (browser bundling), index.html, index.css.
  • Docs: README.md with setup and run instructions.
• Workspace:
  • Add packages/sdk/browser/example to root package.json workspaces.

^{Written by Cursor Bugbot for commit 580a21a. This will update automatically on new commits. Configure here.}

[github-actions]

@launchdarkly/browser size report
This is the brotli compressed size of the ESM build.
Compressed size: 169118 bytes
Compressed size limit: 200000
Uncompressed size: 789399 bytes

[github-actions]

@launchdarkly/js-sdk-common size report
This is the brotli compressed size of the ESM build.
Compressed size: 25394 bytes
Compressed size limit: 26000
Uncompressed size: 124693 bytes

[github-actions]

@launchdarkly/js-client-sdk-common size report
This is the brotli compressed size of the ESM build.
Compressed size: 17636 bytes
Compressed size limit: 20000
Uncompressed size: 90259 bytes

[github-actions]

@launchdarkly/js-client-sdk size report
This is the brotli compressed size of the ESM build.
Compressed size: 22101 bytes
Compressed size limit: 25000
Uncompressed size: 76263 bytes

[cursor]

Bug: UI remains stuck when identify succeeds without flags

The identify call only has a .catch() handler without a .then() for the success case. The UI update relies entirely on the change event firing. However, the SDK only emits a change event when flag values differ from previous values. If no flags are configured in the LaunchDarkly project (or the server returns empty flags), identify succeeds but no change event fires, leaving the UI permanently stuck on "Initializing...". Adding a .then() handler that calls render() would ensure the UI updates after successful identification regardless of whether flags exist.